Conjectures on the Swampland · 1802.08264with Thomas Grimm, Irene Valenzuela 1810.05506with...
Transcript of Conjectures on the Swampland · 1802.08264with Thomas Grimm, Irene Valenzuela 1810.05506with...
Conjectures on the Swampland
Theory ColloquiumCERN, Nov 2018
1705.043281802.08264 with Thomas Grimm, Irene Valenzuela1810.05506 with Hirosi Ooguri, Gary Shiu, Cumrun Vafa
Eran PaltiMax-Planck-Institut für Physik, Munich
Gravity and Quantum Field Theory are notoriously difficult to combine in a single consistent framework that is valid in the ultraviolet
• For ! ∼ #$ → unique theory ?
A critical energy scale is the Planck mass #$ ∼ 10() GeV
• For ! ≪ #$ → anything goes ?
However, from the perspective of low-energy effective theory, ultraviolet problems need not be of concern
Any problem with GR + Maxwell below #$? + = - −/ #$01 +1440 5
0
Within string theory, this apparent freedom, manifests as the Landscape
String Theory (Quantum Gravity)Energy scale
Theory space
Theory space
• Extra dimensions geometry
• Branes, fluxes
• …
A valid question arose:
Does our current knowledge of String Theory imply any non-trivial universal predictions for low-energy theories?
Initial suggestions proposed with the introduction of the Swampland:
• String Theory very much still work-in-progress – final rules not clear• Specific vacua very predictive – can just follow phenomenological approach
The set of self-consistent effective theories that cannot be completed into quantum gravity in the ultraviolet
[ Vafa ’05 ]
The Landscape might be huge, but it is small compared to the Swampland
Set of consistent low-energy effective Quantum Field Theories
String Theory (Quantum Gravity)
Energy scale
Theory space
Theory space
A Swampland Conjecture
String Theory Constructions
Quantum Gravity (Black Holes) New Microscopic Physics
The Swampland programme is about extracting universal predictions from string
theory not as a specific low-energy theory, but as rules governing such theories
Explicit String
Theoretic Origin
“String Inspired”
Prototypical example: Einstein-Maxwell theory in the Swampland
! = # −% &'() +14-( .
(
The Weak Gravity Conjecture
• Must have a charged particle with mass smaller than charge
- / &' ≥ 1• The cutoff scale of the theory (infinite tower of new states) is at
Λ ∼ - &'
Electric WGC
Magnetic WGC
[Arkani-Hamed, Motl, Nicolis, Vafa ’06 ]
Need a charged particle for charged extremal Black Holes to discharge
+-
!
Positively charged black hole
Electric field
The limit ! → 0 is one where the gauge symmetry becomes a global symmetry
Gravity should be the weakest force acting on a particle
m!% Gauge
Gravity
No stable gravitationally bound states
These are indirect, loose arguments: signposts, rather than microscopic physics
Strongest evidence from String Theory [Dine et al ’03; … ; Lee, Lerche, Weigand ‘18]
The Swampland Distance Conjecture
For a massless scalar field !, which undergoes a variation "!, there is an infinite tower of states whose mass scale as "! → ∞ goes as
% ∼ '()*+,-./
for some 0 > 0
It was further proposed that:
• Exponential behavior appears precisely at Δ! ∼ '4 , and 0 ∼ 5(1)• Also holds for fields with a potential 9(!)
[Ooguri, Vafa ’06]
[Baume, EP ‘16; Klaewer, EP ’16]
Prototypical example: compactification on a circle
! → ∞−∞ ← !
&' ∼ )* &++ ∼ ),*
Highly non-trivial evidence this is general in String Theory (for 8 supercharges)
[Ooguri, Vafa ‘06; Cecotti ‘15; Grimm, EP, Valenzuela ‘18; Lee, Lerche, Weigand ’18; Grimm, Li, EP ‘18]
Model-independent general results – highly mathematical
The evidence for fields with a potential has two aspects:
• Simple: the behaviour is set by the field-space metric (the kinetic terms)
!"" # $# % + '(#)
• Highly non-trivial: the potential can itself change appropriately the field-space metric
[EP ‘15; Baume, EP ‘16]
Gravitational backreaction
[Silverstein, Westphal ’08; …]
#Axion monodromy models:
!"" # $# %
# non-canonical
Consider the WGC in the presence of massless scalar fields
m"#
µ
Gauge
Gravity
Scalar
"%#%&'% ≥ )% + +%&'%
ℎ-+ ℎ
ℎ./"# ℎ
ℎ"/0) ℎ
+ = 23)The coupling to scalar fields is
Non-trivial evidence in string theory [EP ‘17; Lee, Lerche, Weigand ‘18]
[EP ‘17]
Imposing that gravity should be the weakest force gives a Scalar WGC
!"# > # [EP ‘17]
This can only hold for large variations of % if we have
# ∼ '("
The distance conjecture can then be thought of as a magnetic version of the Scalar Weak Gravity Conjecture
Can be proven that BPS states satisfy this
Λ ∼ #$%&'()
*+,+()+ ≥ .+ + (1%.)+()+
*+,+()+ ≥ .+
Λ ∼ * ()
Global Symmetries
(1%.)+()+ ≥ .+
Gravity Weakest Force
Electric-Magnetic dualsElectric-Magnetic duals
Asymptotic limit 3 → ∞Asymptotic limit g→ 0
Gravity Weakest Force
We find an inter-related collection of ideas, which hints at underlying physics
?
Proposal: the Swampland conjectures are consequences of the emergent nature of dynamical fields in quantum gravity
[Grimm, EP, Valenzuela ‘18]
See also [Harlow ’15; Heidenreich, Reece, Rudelius ‘17+‘18]
Emergent gauge field toy model CPN
!"∗!" =%&'
Contains a gauge symmetry !" → )"*(,)!", with a gauge field ‘variable’
A ≡ &'21% z3∗4!" − !"4!"∗
The charged scalars develop a mass 67, can integrate them out, and in the IR find an emergent gauge field
ℒ = 4z3∗4z3 + (z3∗4!")(!:∗4!:)
ℒ;< =1
4&;<'?'
The gauge coupling behaves as if it comes purely from 1-loop threshold effects1&;<'
= 1&@A'
+ %12B' log
Λ67
[Witten ‘79]
✗ (Like QED + % massive fields)
Emergent behavior: IR coupling given by integrating down from scale Λ
"#$%&
∼1-loop ( (
More conservative: Scale Λ where reach strong coupling
"#)*&
= "#$%&
−1-loop"#$%&
= "#)*&
+1-loop✗.(0)
[Heidenreich, Reece, Rudelius ‘17+‘18][Harlow ’15; Grimm, EP, Valenzuela ‘18]
Integrating out a tower of states can generate dynamics for gravity/gauge/scalar
!, #, $ !, #, $
Integrate down from a UV scale Λ &'( )*+ = &'( )-. + 0 Λ(
1
2
3
4
5 &'(114#( 4
(
#55 ! 6! (
Fixes the UV cut-off scale as the Species scale Λ = &'0
✗[ …; Dvali ’07 ]
For the gauge coupling we have
The mass scale of the tower
1"#$%
∼ '( ∼ )*%
m%
Magnetic WGC
123
45
1"#$%
= 1"-.%
+01
2 31%65% 67"
Λ91
✗Take equally spaced tower Δ9 ∼ 9, gives ' ∼ ;<
=
>?
9 ∼ "#$)*
“Approximate” Electric WGC
m2m
For scalar field, the 1-loop wavefunction renormalization is
Proper distance
! ∼ #$%&'(Find
)**+, = )**./ +12
34*!2
5
475 89)Λ!2✗
)**+, ∼ ;< 4*!5 ∼ '( =>?
?
5
Distance Conjecture
)+,** 4*!
5 ∼ !5 “Approximate” Scalar WGC
ΔA = B )**+, CD ∼ EF B4*!! CD ∼ −EF log!
For compactifications of type IIB string theory on a Calabi-Yau manifold, have towers of D3 branes wrapping 3-cycles
Integrating them out at 1-loop precisely recovers the behavior of the gauge couplings and scalar fields at any weak-coupling or large distance regime
[Grimm, EP, Valenzuela ‘18]
In String Theory it is sometimes better not to think of fundamental and emergent but rather as a duality
Similar results found for F-Theory on Calabi-Yau[Lee, Lerche, Weigand ‘18]
123
45!
"#$#%
Works for all tested string theory settings (8 supercharges):
Recently, the Swampland de Sitter Conjecture was proposed
!"($) > ' " $ ' ∼ )(1)[Obied, Ooguri, Spodyneiko, Vafa ‘18]
In particular, this forbids de Sitter minima
"($)
$
"($)
$
Cosmological Constant Dynamical Dark Energy(quintessence)
✓✗
Experimentally testable! Euclid, Dark Energy Survey, …
The conjecture was proposed based on a seeming conspiracy against de Sitter in the best-understood string theory constructions
“String Inspired”
No de Sitter vacua Yes de Sitter vacua
The position of the “de Sitter line” is under debate – but it is far enough to the right that one might seriously consider the conjecture
Explicit String Theoretic Origin
Can the de Sitter conjecture be implemented into the coherent picture of the Swampland we have proposed?
It is natural expect some connection between the potential and tower of states
1
2
3
4
5!"#$#%&(!)
In ) = 2 supergravity, the potential is in 1-to-1 correspondence with the gauge coupling matrix: if gauge fields are emergent so must the potential be
[ Ooguri, EP, Shiu, Vafa ‘18]
In de Sitter space the potential can be associated an entropy
!"# = %&' ()* ℋ = ,-[Gibbons, Hawking ’77]
Can be interpreted as the number of states in the Hilbert space
!"# . = 10(.)
de Sitter space has a finite horizon for an observer, of radius ,
[Banks ’00; Witten ‘01]
As we move in field space a tower of ! states becomes light and so the dimension of the Hilbert space of the effective theory increases
The distance conjecture:
We can assign an entropy to the tower below a cut-off scale
"#$%&' ( ∼ !((),-(().
If the tower dominates the Hilbert space, then we can equate the two notions of entropy
10(() ∼ - ( 1 ∼ !((),-((). 0(() ∼ !(()2
1,12.
!(() ∼ 345 6 ∼ 7(1)
Utilising the expression for !(#) from the distance conjecture gives
c ∼ 2()2 − +
,-,# > / -
,-,# =
,-,!
,!,# ∼ ( 2)
2 − + -
This is the de Sitter conjecture
Determining the exponents ) and + amount to the microstates of the tower in quantum gravity – this is a difficult problem
For free fields in a box of size 1 we find ) = 23 and + = 4
5
The argument relied on three assumptions:
• The distance conjecture for fields with a potential
• The states of the tower dominate the Hilbert space
• We can assign an entropy to the potential
• The states of the tower dominate the Hilbert space
This follows at large distances in field space:
i) From the duality with the tower of states
ii) The exponentially large number of states in the tower expect to dominate the Hilbert space
So the assumption holds in any weakly-coupled parametrically controlled regime of string theory
Couplings in String Theory are scalar fields (!", $, %, !&, '&, …)
Weak Coupling ! → 0 Large distance + → ∞
Consider a potential which is away from a minimum so the field is rolling
• We can assign an entropy to the potential
An apparent horizon exists if the universe is accelerating
!"!# ≤ 2 "
The theory is stable on horizon scales (and over a Hubble time) if
!&"!#& ≥ −)(1) "
Finite temperature lifting of mass -.& = 012
0.1 + 4& = 012
0.1 + "
The apparent horizon has area !" as utilized in the computation
An entropy can be associated this horizon using the Covariant Entropy Bound[Fischler, Susskind ’98; Bousso ‘99] Light sheets
sweep out a volume #
Apparent HorizonS(#) ≤ !"
Imposing the assumptions utilized in the derivation we then arrive at a Refined de Sitter Conjecture:
or
We have derived this from the distance conjecture, at any parametrically controlled regime of string theory
!" ≥ $%&
" min(∇,∇-") ≤ − 12345
"
It is natural to conjecture that it holds even away from this regime
[ Ooguri, EP, Shiu, Vafa ‘18]
It is interesting to note that the refinement we were led to also avoids some counter examples to the original conjecture
• The top of the Higgs potential has
!" ∼ 10&'' " min(∇-∇.") ∼ − 1234567
"
• The top of the potential for any axion (including QCD axions and pion)
min(∇-∇.") ∼ − 187 "
The Weak Gravity Conjecture applied to axions gives 9 ≤ ;<
[ Denef, Hebecker, Wrase ‘18]
Summary
• There are a number of existing conjectures about the Swampland, and they form a coherent interlinked framework
• Discussed a proposal for the underlying microscopic physics behind the conjectures: emergence of dynamical fields in quantum gravity
• The de Sitter conjecture can be tied to the distance conjecture in any parametrically controlled regime of string theory